Chapter 9: Wave turbulence theory for capillary water waves

Although many studies have been carried on to understand the Hasselmann-Zakharov weak turbulence equation for capillary waves since its derivation in the 60’s, the question about the existence and uniqueness of solutions to the equation still remains unanswered, due to the complexity of the equation. Our work provides a solution to the problem.

Figure 1: Capillary Waves

First of all, let us review the derivation of the weak (wave) turbulence kinetic equation of capillary waves. We first recall that the law of wave dispersion on the surface of an infinite deep liquid is of the form , in which is the wave number, is the coefficient of the surface tension and is the gravity constant. If we neglect the effect of the gravity, . The water wave system that governs the vibrations of the surface of a liquid, in which the viscosity is omitted, reads

where is the velocity potential, is the deviation of the surface from equilibrium. The axis is directed away from the liquid and we set the pressure to be .

Denote by and (or and ) the Fourier transform of and in and make use the smallness of the nonlinearity, keeping in the Fourier series terms up to second order of smallness of the amplitude of the vibrations

Now, let us define the new variables and , which describe the complex amplitudes of waves

The equation for capillary waves in terms of the new variables then follows

where denotes the positive coefficient of fluid viscosity, and is the collision term, describing pure resonant three-wave interactions. The equation is a three-wave kinetic one, in which the collision operator is of the form

with

We have used the short-hand notation and .

According to the weak turbulence theory, the equation admits nontrivial equilibria , called the Kolmogorov-Zakharov’s spectra: Moreover, such a solution can be interpreted as a universal spectrum in the region of transparency. These solutions are the analogs of the familiar Kolmogorov energy spectrum prediction of hydrodynamic turbulence.

Let us mention that this kinetic wave equation has a very similar structure with the quantum Boltzmann equation that describes the evolution of the excitations in a trapped Bose gas system, in which the temperature of the gas is below the Bose-Einstein condensate transition temperature. The collision operator that describes the interaction between excitations and condensates in the quantum Boltzmann equation reads

with

and , for some positive constants .

In this paper, we develop new techniques, inspired by our recent works on quantum kinetic theory, to give an answer to the question on the global existence and uniqueness of solutions to the equation.